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Oct 24, 2016 - Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality. Ø¡lvaro M. Alhambra,1,* Lluis Masanes,1 Jonathan ...
PHYSICAL REVIEW X 6, 041017 (2016)

Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality Álvaro M. Alhambra,1,* Lluis Masanes,1 Jonathan Oppenheim,1 and Christopher Perry1,2,† 1

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom 2 QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark (Received 16 February 2016; published 24 October 2016)

We investigate the connection between recent results in quantum thermodynamics and fluctuation relations by adopting a fully quantum mechanical description of thermodynamics. By including a work system whose energy is allowed to fluctuate, we derive a set of equalities that all thermodynamical transitions have to satisfy. This extends the condition for maps to be Gibbs preserving to the case of fluctuating work, providing a more general characterization of maps commonly used in the information theoretic approach to thermodynamics. For final states, block diagonal in the energy basis, this set of equalities is a necessary and sufficient condition for a thermodynamical state transition to be possible. The conditions serve as a parent equation that can be used to derive a number of results. These include writing the second law of thermodynamics as an equality featuring a fine-grained notion of the free energy. It also yields a generalization of the Jarzynski fluctuation theorem which holds for arbitrary initial states, and under the most general manipulations allowed by the laws of quantum mechanics. Furthermore, we show that each of these relations can be seen as the quasiclassical limit of three fully quantum identities. This allows us to consider the free energy as an operator, and allows one to obtain more general and fully quantum fluctuation relations from the information theoretic approach to quantum thermodynamics. DOI: 10.1103/PhysRevX.6.041017

Subject Areas: Quantum Physics, Quantum Information, Statistical Physics

I. INTRODUCTION The second law of thermodynamics governs what state transformations are possible regardless of the details of the interactions. As such, it is arguably the law of physics with the broadest applicability, relevant for situations as varied as subatomic collisions, star formation, biological processes, steam engines, molecular motors, and cosmology. For a system that could be placed in contact with a thermal reservoir at temperature T, the second law can be expressed as an inequality of the form hwi ≤ FðρÞ − Fðρ0 Þ;

ð1Þ

where the free P energy is FðρÞ ¼ trHρ − TSðρÞ, the entropy is SðρÞ ¼ − s PðsÞ log PðsÞ, with PðsÞ the probability that the system has energy level jsi, H is the Hamiltonian of the system, and hwi is the average work done by the system when it transitions from ρ to ρ0 . The free energy is a scalar, and can be thought of as an average quantity. Here, we see that by thinking of the free energy as an operator or random * †

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Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

2160-3308=16=6(4)=041017(14)

variable, one can derive a much stronger classical version of the second law, which is an equality, heβðfs0 −fs þwÞ i ¼ 1;

ð2Þ

in terms of a fine-grained free energy, f s ¼ Es þ T log PðsÞ;

ð3Þ

that can be considered as a random variable occurring with probability PðsÞ and whose average value is the ordinary scalar free energy F ¼ hf s i. Here, initial energy levels are given by Es ¼ trjsihsjH, while the energy levels Es0 correspond to the final Hamiltonian H 0 . Although the term −T log PðsÞ is not defined for PðsÞ ¼ 0, all its moments are. We see that this equality version of the second law can be thought of as a simple consequence of a much stronger family of equalities and quantum identities. We also see that if we Taylor expand the exponential in the above equality, we obtain not only the standard inequality version of the second law, but in addition, an infinite set of higherorder inequalities. These can be thought of as corrections to the standard inequality. This second law equality is valid for transitions between any two states as long as the initial state is diagonal in the energy eigenbasis, and when work is considered as the change in energy of some work system or weight. As such, although it is a greatly strengthened form of the second law,

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Published by the American Physical Society

ALHAMBRA, MASANES, OPPENHEIM, and PERRY it is of a more classical nature, reminiscent of the fluctuation theorems of Jarzynski [1] and Crooks [2], where it is required that a measurement is performed on the initial and final state. We are, however, not only able to get a more general classical version of the Jarzynski fluctuation theorem, valid for any initial state, but we are also able to derive two fully quantum identities which reduce in the classical limit to these classical generalizations of the Jarzynski equation and the second law. What is more, even our classical fluctuation theorems are derived from a fully quantum mechanical point of view, and are thus valid for any quantum process. Previous derivations assumed a particular form of Markovian classical trajectories (e.g., assuming a Langevin equation or classical trajectory in the context of classical stochastic thermodynamics) [2–8]. A quantum mechanical derivation of the standard Jarzynski equation has been done in numerous works [9–14], but in these, one is usually limited to initial thermal states, and also must resort to energy measurements on the system in order to define work. A derivation of fluctuation theorems for classical trajectories between arbitrary initial and final states has been performed in the case of erasure and a degenerate Hamiltonian [8]. A derivation with arbitrary initial and final states was also undertaken in Ref. [15] for a family of maps that goes slightly beyond the classical case. Here, we adopt a fully general and quantum mechanical treatment and derive a fully quantum identity, which reduces to the generalized fluctuation relation, heβðw−fs Þ i ¼ Z0S ;

ð4Þ

when the initial state is diagonal in the energy eigenbasis. We call such states, i.e., those that satisfy ½ρS ; HS  ¼ 0, quasiclassical states, and the above fluctuation relation is valid for arbitrary initial and final states of this form and for any quantum thermodynamical process. When the initial state is thermal, we further have e−βfs ¼ ZS for all s, which gives Jarzynski’s equation in its usual form: heβw i ¼

Z0S : ZS

ð5Þ

Our two quantum identities, which reduce to the equality version of the second law and the generalization of the Jarzynski equation valid for arbitrary initial quasiclassical states, can be considered as two independent children of a third, more powerful, quantum identity, trW ½ðJ H0S þHW ΓSW J −1 H S þHW Þð1S ⊗ ρW Þ ¼ 1S ;

ð6Þ

where ρW is the initial state of the weight system, 1S is the identity on the system S of interest, ΓSW is the completely positive trace-preserving map acting on the joint state of system and weight that gives its evolution, and we define,

PHYS. REV. X 6, 041017 (2016) as in Ref. [16] (but with opposite sign convention), J H ðρÞ ¼ eðβ=2ÞH ρeðβ=2ÞH . This parent identity can easily be used to derive a fully quantum, general Jarzynski equation for arbitrary states (Result 3 in Sec. III). When the input is quasiclassical, it reduces to our generalized Jarzynski equation for arbitrary initial quasiclassical states. Likewise, the parent identity gives a fully quantum version of the second law equality, Result 2 in Sec. III, which reduces to the equality version of the second law when the initial state is quasiclassical. Now, it is natural to ask what the parent identity, Eq. (6), reduces to for quasiclassical states. While Eq. (6) must necessarily be fulfilled by all thermodynamical processes on quantum states, on quasiclassical states it leads to the following necessary and sufficient condition for transition probabilities to be realizable through thermal processes: X Pðs0 ; wjsÞeβðEs0 −Es þwÞ ¼ 1;

ð7Þ

s;w

for all s0 , where Pðs0 ; wjsÞ is the conditional probability of the final state having energy levels Es0 , and work w being done by the system, given that the initial state had energy level Es . This turns out to be the extension of an important equation from the resource theoretic approach to quantum thermodynamics, which finds its origin in ideas from quantum information theory [17–45]. An overarching idea behind the information theoretic approach is to precisely define what one means by thermodynamics, and thus consider which possible interactions are allowed between a system, a heat bath, and a work storage device, while systematically accounting for all possible resources used in the process. This leads to a definition of thermodynamics known as thermal operations (TO) [17,22,46]. This, and its catalytic version [24], represent the most an experimenter can possibly do when manipulating a system without access to a reservoir of coherence (although one can easily include a coherence reservoir as an ancilla as in Refs. [21,28,32]). It is thus the appropriate class of operations for deriving limitations such as a second law. However, they can be realized by very coarse-grained control of the system, and thus also represent achievable thermodynamical operations [42]. They also include the allowed class of operations considered in fluctuation theorems, which include arbitrary unitaries on system and bath. We explain this inclusion in the Appendix. Thermal operations are thus broad enough to encompass commonly considered definitions of thermodynamics (see Ref. [21] for further discussion on this point), including those used in the context of fluctuation relations. Equation (7) turns out to completely characterize thermodynamics in the case of fluctuating work. In information theory, an important class of maps are those that satisfy the doubly stochastic condition, i.e., preservation of the maximally mixed state. In thermodynamics,

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FLUCTUATING WORK: FROM QUANTUM …

PHYS. REV. X 6, 041017 (2016)

when there is no work system, any operation must instead preserve the Gibbs state. Equation (7) is an extension of this condition to the case where there is a work system that is allowed to fluctuate. Taking Eq. (7) with w ¼ T logðZS =Z0S Þ gives the Gibbs-preserving condition derived in Refs. [17,46]. We show that Eq. (7) provides a necessary and sufficient condition for thermodynamical transitions between states which are diagonal in the energy eigenbasis, and as a result is a necessary and sufficient condition for work fluctuations. Using a generalization of a theorem of Hardy, Littlewood, and Polya [47], the condition of Gibbs preservation was shown in Ref. [48] to be equivalent to the set of thermodynamical second laws that have recently been proven to be a necessary and sufficient condition for quantum thermodynamical state transformations [22] (cf. Ref. [47]), the so-called thermomajorization criteria [22,49]. The latter are conditions on the initial probabilities PðsÞ and final probabilities Pðs0 Þ under which one state can be transformed into another. Previously, in the resource theory approach, the work storage system had to be taken to be part of the system of interest [22]. Using this technique, one can compute the minimal amount of deterministic work required to make a state transformation [22] using thermomajorization. One can also consider fluctuating or probabilistic work from the resource theoretic perspective, but thus far, only average work has been computed [21,26]. Unresolved, thus far, has been the question of how the information theoretic paradigm fits in with the fluctuation theorem approach. Some further insights have been obtained in attempting to link the information theoretic approach with the fluctuation theorem approach [36,50,51]; however, how the two paradigms fit together has remained an open question. Here, we see that one can incorporate fluctuating work explicitly in the resource theoretic approach through Eqs. (6) and (7). These serve to bring the field of fluctuation theorems fully into the domain of the information and resource theoretic approach. This is possible because the class of operations that are allowed in the fluctuation theorem paradigm lies within thermal operations. The latter approach is also able to incorporate not only fluctuations of work but also of states [52,53], and here we aim to extend its use to further physically motivated situations. Finally, it is interesting to compare the power of the relations presented here with the Jarzynski and Crooks’s relations. We do this for one of the simplest examples, the process of Landauer erasure [54], where a bit in an unknown state is erased to the 0 state. Since the initial state is thermal, one can apply the Jarzynski equality in its standard form. However, even in this simple case, we find that the new equalities proven here give more information than the standard Jarzynski and Crooks, in part because one has an independent equality for each possible final energy state. One finds a number of additional insights. Namely, (i) that one needs very large work fluctuations that grow

exponentially as the probability that the erasure fails decreases—the more perfect we demand our erasure to be, the larger the work fluctuations, (ii) it is impossible to even probabilistically extract work in a perfect erasure process, and (iii) that the optimal average work cost for perfect erasure of T log 2 is achieved only when the work fluctuations associated with successful erasure tend to zero. While these facts are known for protocols that are thermodynamically reversible, we know of no proof that they hold for arbitrary protocols. This simple application is discussed in Sec. VIII. The remainder of this paper is structured as follows: in Sec. II, we define what we consider to be thermodynamics— namely, the set of thermal operations in the presence of fluctuating work. This involves three simple conditions on the type of operations that can be performed, and we find some general constraints they need to obey. In Sec. III, we introduce the three fully quantum thermodynamic identities and prove them. In Sec. IV, we show that in the case of states that are diagonal in their energy eigenbasis, these quantum identities each reduce to the equality version of the second law, a generalization of the Jarzynski equation, and the extension of the Gibbs-preservation condition to the case of fluctuating work. In Sec. V, we discuss the implications of our results on determining conditions for state transformations to be possible. In Sec. VI, we show that in the case of the initial state being diagonal in the energy basis and the final state being arbitrary, the quantum identities reduce to constraints on the expectation values of certain operators with a clear physical interpretation. Recently, a fully quantum Crooks-type identity was derived in Ref. [16]. This gives a constraint on the quantum state of the weight depending on both the evolution and the time-reversed evolution. As our constraints are on both the system and weight, the two results appear to complement each other without overlap. We relate the two results by proving a quantum analog of the Crooks relation, of similar form to that in Ref. [16] but applying not just to the weight but to the system and weight. This is done in Sec. VII. Some of the other results in Ref. [16] can be derived in our framework as well. II. THERMAL OPERATIONS WITH FLUCTUATING WORK First, let us characterize the type of process or operation that we consider and show that they are suitably general and implementable to encompass what is commonly considered to be thermodynamics. Our setting consists of a system with Hamiltonian HS , a bath with Hamiltonian HB initially in the −βHB , and an ideal weight with thermal state ρB ¼ ð1=Z R B Þe Hamiltonian HW ¼ R dxxjxihxj, where the orthonormal basis fjxi; ∀x ∈ Rg represents the position of the weight. The operations we consider allow for the Hamiltonian to change, as we see in Sec. II A. Any joint transformation of system, bath, and weight is represented by a completely

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ALHAMBRA, MASANES, OPPENHEIM, and PERRY positive trace-preserving (CPTP) map ΓSBW . We consider only maps ΓSBW satisfying the following conditions. (i) Unitary on all systems.—It has an (CPTP) inverse Γ−1 SBW , which implies unitarity: ΓSBW ðρSBW Þ ¼ UρSBW U † . (ii) Energy conservation.—The unitary commutes with the total Hamiltonian: ½U; HS þ H B þ HW  ¼ 0. (iii) Independence of the “position” of the weight.— The unitary commutes with the translations on the weight: ½U; ΔW  ¼ 0. Here, ΔW is the generator of the translations in the weight and canonically conjugate to the position of the weight HW ; that is, ½ΔW ; H W  ¼ i. Note that these constraints allow for processes that exploit the coherence of the weight, as in Refs. [26,55]. We expand on this in the Appendix, where we show that such coherence can allow us to implement arbitrary unitaries on system and bath. Both unitarity and energy conservation are fundamental laws of nature, so imposing them is a necessary assumption. Any process that appears to violate energy conservation in the above sense is, in fact, energy conserving; one is merely tracing out or ignoring a system that is taking or giving up energy. For example, turning on an interaction Hamiltonian between system, bath, and weight can be done via a coherent ancilla, as demonstrated in Ref. [21]. One can generate other couplings between the three systems via the unitary U. Imposing energy conservation on the systems considered thus ensures that all sources of energy are properly accounted for. Note that while we require that the total process is unitary on the systems, weight, and bath, the operation on system and weight alone will not usually be unitary. The last condition, independence of the weight position, implies that the reduced map on system and bath ΓSB is a mixture of unitaries (Result 1 in Ref. [55]). Hence, the transformation can never decrease the entropy of system and bath, which guarantees that the weight is not used as a resource or as an entropy sink. Independence of the position of the weight can be thought of as a definition of work [26] and is used in both the information theoretic and fluctuation theorem approaches. In the latter case, the assumption is implicit, since the amount of work is taken to be the difference in energy between the initial and final system and bath. In other words, work is taken to be a change in energy of either the work system (explicit) or a change in energy of the system-bath (implicitly). Conservation of energy ensures that the implicit and explicit paradigms are equivalent. Work then is the change in energy of the work system, and does not depend on how much energy is currently stored there; hence, the unitary must commute with its translations. In the Appendix, we discuss the connection between different paradigms in more detail and, in particular, show that thermal operations is sufficiently general to include the paradigms typically considered in the context of fluctuation relations.

PHYS. REV. X 6, 041017 (2016) A. Thermal operations with nonconstant Hamiltonian Thermal operations are general enough to include the case where the initial Hamiltonian of the system HS is different than the final one H0S . This is done by including an additional qubit system X, which plays the role of a switch (as in Ref. [22]). Now the total Hamiltonian is H ¼ HS ⊗ j0iX h0j þ H 0S ⊗ j1iX h1j þ HB þ HW ;

ð8Þ

and energy conservation reads ½V; H ¼ 0, where V is the global unitary when we include the switch. We impose that the initial state of the switch is j0iX and the global unitary V performs the switching, VðρSBW ⊗ j0iX h0jÞV † ¼ ρ0SBW ⊗ j1iX h1j;

ð9Þ

for any ρSBW. This implies ~ ⊗ j0iX h1j; V ¼ U ⊗ j1iX h0j þ U

ð10Þ

~ are unitaries on system, bath, and weight. where U and U The condition ½V; H ¼ 0 implies UðHS þ HB þ H W Þ ¼ ðH 0S þ HB þ HW ÞU:

ð11Þ

Therefore, the reduced map on system, bath, and weight can be written as ΓSBW ðρSBW Þ ¼ UρSBW U† ;

ð12Þ

where the unitary U does not necessarily commute with HS þ HB þ H W nor H0S þ H B þ HW but satisfies Eq. (11). III. IDENTITIES FOR QUANTUM THERMAL OPERATIONS In this section, we derive some fully quantum equalities for thermal operations with fluctuating work. In the next section, we provide the physical meaning of these equalities. Thus far, from the information theoretic perspective, some quantum constraints on state transformations are known, i.e., constraints on transformations of the coherences over energy levels [24,28–32], but none of these constraints apply in the case of fluctuating work. On the other hand, in the fluctuation theorem approach, no quantum relations are known—one always assumes that the initial and final states are measured in the energy eigenbasis; thus, one is considering only transitions between quasiclassical states. In what follows we are mostly interested in the joint dynamics of system and weight, which is described by the CPTP map:

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FLUCTUATING WORK: FROM QUANTUM …     e−βHB U† : ΓSW ðρSW Þ ¼ trB U ρSW ⊗ ZB

PHYS. REV. X 6, 041017 (2016) ð13Þ

It is convenient to define the CP (but not TP) map, J H ðρÞ ¼ eðβ=2ÞH ρeðβ=2ÞH ;

ð14Þ

whose inverse is −ðβ=2ÞH J −1 ρe−ðβ=2ÞH : H ðρÞ ¼ e

ð15Þ

In the case where the initial state ρS is not full rank, we can take the limit of a full-rank state. Now, applying J T ln ρ0S and taking the trace over S on both sides of Eq. (18), we obtain the following. Result 2. (Quantum second law equality).—If ΓSW is a thermal operation, then for every pair of initial states ρS , ρW , we have −1 trSW ½ðJ T ln ρ0S J H0S þHW ΓSW J −1 H S þH W J T ln ρS ÞðρS ⊗ ρW Þ ¼ 1;

ð19Þ

Using Eqs. (11) and (12), we obtain where

ðJ H0S þHW ΓSW J −1 H S þH W Þð1SW Þ   −βðH þH þH Þ  S B W e U† ¼ J H0S þHW trB U ZB   −βðH0 þH þH Þ  B W S e ¼ J H0S þHW trB ZB

ρ0S ¼ trW ½ΓSW ðρS ⊗ ρW Þ

0

¼ J H0S þHW ðe−βðHS þHW Þ Þ ¼ 1SW :

ð16Þ

As we mention in the previous section, it was proven in Ref. [55] that the condition ½U; ΔW  ¼ 0 implies trW ½Uð1SB ⊗ ρW ÞU †  ¼ 1SB for any state ρW . Proceeding similarly as in Eq. (16), we obtain trW ðJ H0S þHW ΓSW J −1 H S þH W Þð1S ⊗ ρW Þ   1 −1 † tr ½UJ HS þHB þHW ð1SB ⊗ ρW ÞU  ¼ trW J H0S þHW ZB B   1 † ¼ trW J H0S þHW trB ½J −1 ðU1 ⊗ ρ U Þ SB W H 0S þH B þH W ZB  −βH  e B ¼ trBW Uð1SB ⊗ ρW ÞU † ZB  −βH  e B ¼ trB 1 ZB SB ¼ 1S :

ð17Þ

We thus have the following. Result 1. (Quantum Gibbs stochastic).—If ΓSW is a thermal operation, then trW ½ðJ H0S þHW ΓSW J −1 H S þHW Þð1S ⊗ ρW Þ ¼ 1S

ð18Þ

for any initial state of the weight ρW . This is a quantum extension of the Gibbs-preservation condition presented in Refs. [17,46]. The result generalizes that in Refs. [17,46], not only because it includes work, but also because it is fully quantum. The details of the quasiclassical generalization to the case of fluctuating work are provided in Sec. IV. Next, we use the identities J −1 T ln ρ ðρÞ ¼ 1 and trS ½J T ln ρ ð1Þ ¼ 1, which hold for any full-rank state ρ.

ð20Þ

is the final state of the system. The above result is a quantum generalization of the second law equality, which we describe in Sec. IV. Now, applying J −1 H 0S and taking the trace over S on both sides of Eq. (18), we obtain a quantum generalization of the Jarzynski inequality for general initial states as follows. Result 3. (Quantum Jarzynski equality).—If ΓSW is a thermal operation, then −1 0 trSW ½ðJ HW ΓSW J −1 H S þH W J T ln ρS ÞðρS ⊗ ρW Þ ¼ Z S

ð21Þ

for every pair of initial states ρS , ρW . IV. IDENTITIES FOR CLASSICAL THERMAL OPERATIONS We now go from the fully quantum identities to ones that are applicable for quasiclassical states (i.e., those considered in fluctuation theorems). We thus consider the case where there is an eigenbasis jsi for H S and an eigenbasis js0 i for H0S such that X ΓSW ðjsihsj ⊗ j0ih0jÞ ¼ Pðs0 ; wÞjs0 ihs0 j ⊗ jwihwj; s0 ;w

ð22Þ where jwi are eigenstates of HW . Note that when HS or H0S are degenerate, they could have other eigenbases not satisfying the above. We say that ΓSW is a process that acts on quasiclassical states. Also, the “independence of the position of the weight” allows us to choose its initial state to be j0i without loss of generality. If we denote by Es and Es0 the eigenvalues corresponding to jsi and js0 i, then we can write J HS ðjsihsjÞ ¼ eβEs jsihsj and J H0S ðjs0 ihs0 jÞ ¼ eβEs0 js0 ihs0 j. When Eq. (22) holds, we can represent the thermal operation ΓSW by the stochastic matrix Pðs0 ; wjsÞ ¼ tr½js0 ihs0 j ⊗ jwihwjΓSW ðjsihsj ⊗ j0ih0jÞ: ð23Þ In such a case we have the following.

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ALHAMBRA, MASANES, OPPENHEIM, and PERRY Result 4. (Classical Gibbs stochastic).—Pðs0 ; wjsÞ is a thermal operation mapping quasiclassical states to quasiclassical states if and only if X Pðs0 ; wjsÞeβðEs0 −Es þwÞ ¼ 1

ð24Þ

PHYS. REV. X 6, 041017 (2016) change, setting w ¼ 0 in Result 4 reproduces the aforementioned result. We discuss the implications of this condition on state transformations in the next section. In a similar fashion to the previous section, we can write the quasiclassical version of Result 2 as X

s;w 0

for all s . Proof.—The proof of the only if direction follows simply by writing Result 1 in terms of the matrix of Eq. (23). The if direction is proven as follows. Let us consider a bath with infinite volume in a thermal state at inverse temperature β. Without loss of generality, the energy origin of the bath can be chosen such that hEiβ ¼ 0. This and the fact that its heat capacity is infinite (due to the infinite volume) imply that the density of states ΩðEÞ is proportional to eβE . Because of energy conservation and invariance of the position of the weight, the joint map of system, bath, and weight can be characterized by a map on system and bath π: ðs; bÞ → ðs0 ; b0 Þ, where ðs; bÞ and ðs0 ; b0 Þ label pairs of system and bath energy levels. We construct the map π from the given Pðs0 ; wjsÞ in the following way. When the system makes the transition s → s0 , a fraction Pðs0 ; w ¼ E − E0 þ Es − Es0 jsÞ of the bath states with energy E are mapped to bath states with energy E 0, for all values of E. Using the fact that the number of states with energy E is ΩðEÞ ¼ AeβE (for some constant A), we now show that π is a permutation. The number of (final) states in the set fðs0 ; b0 Þ∶E b0 ¼ E 0 g is ΩðE 0 Þ. And the number of (initial) states ðs; bÞ that are mapped to this set is X Pðs0 ; w ¼ E − E0 þ Es − Es0 jsÞΩðEÞ s;E

Pðs0 ; wjsÞeβðfs0 −fs þwÞ PðsÞ ¼ 1;

ð25Þ

s0 ;s;w

where we define the fine-grained free energies 1 f s ¼ Es þ ln PðsÞ; β

ð26Þ

1 f s0 ¼ Es0 þ ln Pðs0 Þ: β

ð27Þ

In a more compact form, we have the following. Result 5. (Classical second law equality).—A process on quasiclassical states that acts unitarily on the total system, conserves energy, and is independent of the position of the weight satisfies heβðfs0 −fs þwÞ i ¼ 1:

ð28Þ

This result follows simply by using Eq. (23) in Result 2 or directly from Result 4. Because of the convexity of the exponential, this equality implies the standard second law: hf s0 − f s þ wi ≤ 0:

ð29Þ

But Eq. (28) is stronger, since it implies the following infinite list of inequalities:

X 0 ¼ Pðs0 ; wjsÞAeβðE s0 −Es þwþE Þ s;w

N X βk

¼ ΩðE 0 Þ;

k¼1

where in the last line we use Eq. (24). Therefore, it is possible to construct a permutation with the mentioned requirements. ▪ Note that Result 4 gives a necessary and sufficient condition that thermal operations with a fluctuating weight must satisfy for transformations between quasiclassical states, while the fully quantum Result 1 is a necessary condition. This last point can be seen by considering an operation that is Gibbs preserving on the system and acts as 1W on the weight. This clearly satisfies Eq. (18), yet since Gibbs-preserving operations are a larger class of operations than thermal operations [56], it need not be a thermal operation. The above is an extension of the Gibbs preservation condition [17,46] to the case where thermodynamical work is included. When the Hamiltonian of the system does not

k!

hðf s0 − f s þ wÞk i ≤ 0;

ð30Þ

where N can be any odd number. Note that Eq. (29) is the N ¼ 1 case. One can think of Eq. (30) as providing higherorder corrections to the standard second law inequality. All the other inequalities have information about the joint fluctuations of f s , f s0 , and w. To prove Eq. (30) we just note that the residue of the Taylor expansion of the exponential function to any odd order is always negative. Next, we proceed as in Result 5, and obtain the classical version of Result 3. Once again, this can be done either by substituting Eq. (23) into Result 3 or proceeding directly from Result 4. Result 6. (Classical Jarzynski equality).—A process on quasiclassical states that acts unitarily on the total system, conserves energy, and is independent of the position of the weight satisfies

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FLUCTUATING WORK: FROM QUANTUM … heβðw−fs Þ i ¼ Z0S :

PHYS. REV. X 6, 041017 (2016)

Note that this version of the Jarzynski equation is valid for any initial state of the system, encoded in the finegrained free energy f s. For the particular case where the initial state is thermal, we have e−βfs ¼ ZS for all s, which implies the standard Jarzynski equality: heβw i ¼

Z0S : ZS

- (E1+ 1) e

ð31Þ

- (E2+ 2) e

- (E2+ 2) e

- (E1+ 1) e

1

ð32Þ

V. IMPLICATIONS FOR CONDITIONS ON STATE TRANSFORMATIONS

, ,

Result 4, the extension of the Gibbs-preserving condition to the case of fluctuating work, is a restriction on what maps are possible in thermodynamics. The Gibbs-preserving condition is likewise a generalization of the condition for a map to be bistochastic (i.e., preserve the maximally mixed state). This is recovered when the initial and final Hamiltonians of the system are trivial, HS ¼ HS0 ¼ 0, and we set w ¼ 0 in Result 4. For the case of bistochastic maps Λ, the condition on the map is equivalent to a condition on what state transformations are possible. Namely, for two states ρ and ρ0 , ρ0 ¼ ΛðρÞ if and only if ρ0 is majorized by ρ [47]. The majorization condition is as follows. For eigenvalues of ρ and ρ0 arranged in nonincreasing order and denoted byP λs, λ0s , weP say that ρ0 is majorized by ρ if and only if ks¼1 λs ≥ ks¼1 λ0s ∀k. For nontrivial Hamiltonians such that HS ¼ HS0 (but still setting w ¼ 0), thermal operations preserve the associated Gibbs state rather than the maximally mixed state. For such sets of allowed operations, thermomajorization provides a set of conditions that relate initial states to achievable final states (see Fig. 1). These can be considered as a refinement of the second law of thermodynamics, since they constrain the states to which some initial state can evolve under the laws of thermodynamics. In the case of fluctuating work (no longer requiring that w ¼ 0), one can now ask whether it is possible to relate the condition on allowed maps given by Result 4 to a condition akin to thermomajorization on the achievable states and work distributions. A simple, and fairly common case is where the values of work that occur are labeled by s and s0 and of the form wss0 ¼ αs0 − γ s . These can be readily related to the Gibbs-preservation condition of Refs. [17,46] (note that processes such as level transformations as discussed in the Appendix are examples of such transformations). For such work distributions, Pðs0 ; wss0 jsÞ is a thermal operation if and only if Pðs0 jsÞ ≡ Pðs0 ; wss0 jsÞ satisfy the Gibbspreservation condition of Refs. [17,46] but with the energy levels of the initial and final systems redefined so that the initial energy levels are Es þ γ s and the final are Es0 þ αs0 . Determining whether it is possible to convert a state ρ into a

s s

0 0

e-

E2

e-

E1

ZS

P FIG. 1. Given a P system in state ρ ¼ ns¼1 ps jsihsj with Hamiltonian HS ¼ ns¼1 Es jsihsj, its thermomajorization diagram (see Ref. [22] for more details) is formed by first relabeling the pairs of occupation probabilities and energy levels so that βE1 βEn 2 pP ≥ p2 eβE and then plotting the points 1e Pk≥    ≥n pn e k −βEs f s¼1 e ; s¼1 ps gk¼1 , joining them together to form a concave curve. Here, we show examples for a qubit. In the absence of a work storage system, ðρ; HS Þ can be transformed into ðσ; HS Þ using a thermal operation if and only if the curve associated with ρ is never below that of σ. In this example, the curve of ρ crosses that of σ, so the transformation is not possible. When all values in a work distribution have the form wss0 ¼ αs0 − γ s , the existence of a thermal operation mapping a quasiclassical state ρ to quasiclassical state σ while producing such a work distribution canPbe determined by considering P the curves associated with ðρ; ns¼1 ðEs þ γ s ÞjsihsjÞ and ðσ; ns¼1 ðEs þ αs ÞjsihsjÞ. In this example, the curve associated with ρ and fγ s g lies above that of σ and fαs g, so the transformation from ρ to σ is possible with respect to this work distribution. By adjusting γ s and αs so that both curves are straight lines that overlap, one can make the average work of the transformation equal to the change in free energy, and the transformation becomes reversible.

state σ while extracting work of the form wss0 ¼ αs0 − γ s can be done using the thermomajorization diagrams introduced in Ref. [22], as shown in Fig. 1. Indeed, when αs0 ¼ −Es0 and γ s ¼ −Es , the problem reduces to the question of whether ρ majorizes σ. VI. CLASSICAL-QUANTUM IDENTITIES In classical physics no problem arises from writing joint expectations of observables for the initial and final states of an evolution. For example, this is done in Results 4–6. In general, quantum theory does not allow for this, because a measurement on the initial state will disturb it, and then it will no longer be the initial state. However, in the case

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ALHAMBRA, MASANES, OPPENHEIM, and PERRY where the measurement is nondisturbing on the initial state, the joint expectation is well defined, independently of the measurement on the final state. In what follows, we analyze this case by imposing that both the system and weight are initially quasiclassical. We do not impose any constraint on the final state, but define its dephased version by Z 0 0 Δ0 ½ρ0S  ¼ dte−iHS t ρ0S eiHS t : ð33Þ This dephasing CPTP map projects ρ0S onto the subspace of Hermitian P matrices that commute with H 0S . If ρS ¼ s PðsÞjsihsj is the spectral decomposition of the initial state, and jxi is an eigenstate of HW , then −1 ðJ −1 H S þH W J T ln ρS Þðjsihsj

⊗ jxihxjÞ

¼ e−βðEs þT ln PðsÞþxÞ ðjsihsj ⊗ jxihxjÞ:

ð34Þ

The following definitions of the free-energy operator are used below: FS ¼ H S þ T ln Δ½ρS ;

ð35Þ

F0S ¼ H 0S þ T ln Δ0 ½ρ0S :

ð36Þ

If in the derivation of Result 2 we multiply Eq. (18) by J T ln Δρ0S instead of J T ln ρ0S , we obtain trSW ½ðJ F0S þHW ΓSW J −1 FS þH W ÞðρS ⊗ ρW Þ ¼ 1;

PHYS. REV. X 6, 041017 (2016) proven in Ref. [16] but on the weight and system. We also derive a classical version directly from our generalized Gibbs-stochastic condition and without the need to assume microreversibility. In relation to the map defined in Eq. (13), we can also define the associated backwards CPTP map associated:     e−βHB † ΘSW ðρSW Þ ¼ trB U ρSW ⊗ U : ð41Þ ZB Like any CP map, this can be written in Kraus form: X ΘSW ðρSW Þ ¼ Ak ρSW A†k : ð42Þ k

The dual of a map is defined as X A†k ρSW Ak : ΘSW ðρSW Þ ¼ A bit of algebra shows that ΘSW ðρSW Þ

 J H0S þHW ΓSW J −1=2 H S þHW ¼ ΘSW :

Pback ðs; −wjs0 Þ ¼ Pðs0 ; wjsÞeβðEs0 −Es þwÞ :

ð38Þ

0

heβFS eβW e−βFS i ¼ 1:

ð39Þ

In the same way, we have the following. Result 8. (Classical-quantum Jarzynski equation): heβðW−FS Þ i ¼ Z0S :

ð40Þ

VII. QUANTUM CROOKS RELATION Here, we use our techniques to prove a fully quantum version of the Crooks relation, which is related to that

ð44Þ

maps, ð45Þ

This shows that the dual map is analogous to the transpose map that appears in various results of quantum information theory [57,58]. Note that using the classical version of generalized Gibbs stochasity, Result 4, we can define the map

s

or, equivalently, Result 7. Result 7. (Classical-quantum second law equality).— Consider a process that acts unitarily on the total system, conserves energy, and is independent of the position of the weight. If the initial states of system and weight commute with the corresponding Hamiltonians, then

 −βH  e B † ¼ trB UðρSW ⊗ 1B ÞU ; ZB

from which Result 9 follows. Result 9.—The forward and backward respectively, ΓSW and ΘSW , are related via

ð37Þ

where we use that Δ½ρS  ¼ ρS . Again, independence from the position of the weight allows us to choose ρW ¼ j0ih0j. This enables us to write the above equality as X 0 e−βfs PðsÞtrSW ½eβðFS þHW Þ ΓSW ðjsihsj ⊗ j0ih0jÞ ¼ 1;

ð43Þ

k

ð46Þ

One can check that the constraint in Eq. (24) applied to Pðs0 ; wjsÞ is equivalent to the normalization of Pback ðs; −wjs0 Þ, and the normalization of Pðs0 ; wjsÞ is equivalent to the constraint in Eq. (24) applied to Pback ðs; −wjs0 Þ. This constraint implies that Pback ðs; wjs0 Þ is a thermal operation; hence, there is a global unitary generating this transformation. It can also be seen that one can use the unitary that is the inverse of the one that generates Pðs0 ; wjsÞ (although other unitaries may also generate the same dynamics on system and weight). Pback ðs; −wjs0 Þ is thus the microscopic reverse of Pðs0 ; wjsÞ. Indeed, by defining the probability of obtaining work w in going from energy level s to s0 when the initial state is thermal by pforward ðw; s0 ; sÞ ¼ Pðs0 ; wjsÞe−βEs =ZS for the forward process and pback ð−w; s0 ; sÞ ¼ Pback ðs0 ; −wjsÞe−βEs0 =Z0S for the reverse, we obtain a Crooks relation,

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FLUCTUATING WORK: FROM QUANTUM … 0 pforward ðw; s; s0 Þ −βw ZS ; ¼ e pback ð−w; s; s0 Þ ZS

PHYS. REV. X 6, 041017 (2016) w0

ð47Þ

Log 2

〈w〉

without needing to assume microreversibility, which is the starting assumption of Refs. [2,59]. One can take pback ð−w; s; s0 Þ to the rhs of Eq. (47) and then sum over s and/or s0 to obtain the more standard Crooks relation, pforward ðwÞ Z0 ¼ e−βw S ; pback ð−wÞ ZS

Log 2

ð48Þ

but Eq. (47) is clearly stronger. VIII. CONCLUSION In this paper, we consider thermodynamical operations between a system, a thermal bath, and a weight from which one can extract work in a probabilistic way. From a small set of physically motivated assumptions, one can show that these operations obey an identity on arbitrary states from which a number of new, or more general, equalities can easily be found. The equalities are both of a fully quantum and of a classical nature. One of these, the second law as an equality, is of a much stronger form than the standard second law. For example, the saturation of the second law inequality, hf s0 − f s þ wi ¼ 0;

ð49Þ

implies w ¼ f s − f s0

for all s; s0 :

ð50Þ

This regime is called thermodynamically reversible and provides the optimal consumption or extraction of work when we take its average hwi as the figure of merit. Outside of the thermodynamically reversible regime, violations of f s0 − f s þ w ≤ 0

ð51Þ

for individual realizations of the process ðs; s0 ; wÞ can occur. Defining the excess random variable, v ¼ f s0 − f s þ w, allows us to write Eq. (28) as heβv i ¼ 1:

ð52Þ

Recalling that the exponential function gives more weight to the positive fluctuations as compared with the negative ones, we conclude that, outside of the thermodynamically reversible regime, the negative fluctuations of v must be larger and/or more frequent than the positive ones. In other words, the violation of the second law is more rare than its satisfaction. This asymmetry is also articulated by the infinite list of bounds for the moments of v given in Eq. (30).

FIG. 2. As a simple example of the second law equality, one can think of single qubit erasure. In the limit of perfect erasure, the second law equality reads in this case eβw0 þ eβw1 ¼ 1, and the average work spent is hwi ¼ 12 ðw0 þ w1 Þ. Here, we show the trade-off between w0 and hwi for such perfect erasure. The optimal work value for erasure is the usual Landauer cost at w0 ¼ w1 ¼ −T log 2. As seen in Eq. (53), perfect or near-perfect erasure requires the work cost to fluctuate arbitrarily.

Note that the Gibbs-stochastic condition of Eq. (7) gives more information than the Jarzynski equation or second law equality, as the number of constraints it imposes is given by the dimension of the final system. In fact, each condition can be thought of as a separate second law equality—a situation that parallels the fact that one has many second laws for individual systems [22,24]. This is related to the fact that in the case with no weight, Gibbs stochasity is equivalent to these additional second laws given by thermomajorization [48]. As a concrete and simple example of these conditions, let us take the case of Landauer erasure [54]. We consider a qubit with HS ¼ 0 that is initially in the maximally mixed state and which we want to map to the j0i state. Recalling that a positive work value represents a yield, while a negative work value is a cost, we consider a process such that −w0 is the work cost when erasing j0i → j0i, and −w1 the work cost if the transition j1i → j0i occurs. We allow for an imperfect process and imagine that this erasure process happens with probability 1 − ϵ, while with probability ϵ we have an error and either j0i → j1i with work yield w0 or j1i → j1i with work yield w1 . We call such a process deterministic, because w is determined by the particular transition. For this scenario, the the generalized Gibbs-stochastic condition, Eq. (7), gives two conditions: eβwo þ eβw1 ¼ 1=ð1 − ϵÞ;

ð53Þ

eβwo þ eβw1 ¼ 1=ϵ:

ð54Þ

We immediately see that to obtain perfect erasure, ϵ → 0, then when the erasure fails there must be work fluctuations that scale like −T log ϵ. Such a work gain happens rarely,

041017-9

ALHAMBRA, MASANES, OPPENHEIM, and PERRY but precludes perfect erasure. It is related to the third law proven in Ref. [55] and is discussed in detail in Ref. [60]. In the limit of perfect erasure, we illustrate the work fluctuations in Fig. 2. We easily see that the minimal average work cost of erasure T log 2 is obtained when the work fluctuations associated with successful erasure are minimal. We also see that no work, not even probabilistically, can be obtained in such a deterministic process. Since Eq. (7) is not only necessary but also sufficient, we can achieve these work distributions just through the very simple operations described in Ref. [42]. Through this example, one sees that the identities proven here can lead to new insights in thermodynamics, particularly with respect to work fluctuations and their quantum aspects. ACKNOWLEDGMENTS We would like to thank Gavin Crooks for interesting comments on the first version of this paper. We are grateful to the EPSRC and Royal Society for support. C. P. acknowledge financial support from the European Research Council (ERC Grant Agreement No. 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). A. M. A. acknowledges support from the FQXi. APPENDIX: IMPLEMENTING ARBITRARY UNITARIES ON SYSTEM AND BATH

PHYS. REV. X 6, 041017 (2016) consider only unitaries that do not create coherence. Nonetheless, for greater generality, we describe how to implement an arbitrary unitary. To do this, we show that given the three fundamental constraints we impose on our allowed operations in Sec. II, (unitarity, energy conservation, and independence on the state of the weight), we can give a characterization of the unitary transformation on system and bath. As a consequence, we find that arbitrary unitaries on system and bath can be implemented, and we then show how to obtain the distribution on the work system. In what follows, it is useful to denote the eigenvectors of theR generator of the translations on the weight ΔW ¼ dttjtihtj by jti. The following result shows that when implementing an arbitrary unitary, the dynamics of the weight is fully constrained, and that the remaining freedom is implicitly characterized by a system-bath unitary. Lemma 1.—A map ΓSBW obeys the three constraints of Sec. II (unitarity, energy conservation, and independence on the state of the weight) if and only if there is an arbitrary system-bath unitary V SB such that the global unitary on system, bath, and weight can be written as 0

U SBW ¼ eiðHS þHB Þ⊗ΔW ðV SB ⊗ 1W Þe−iðHS þHB Þ⊗ΔW Z ¼ dtASB ðtÞ ⊗ jtihtj; where we define the family of unitaries:

Here, we show how more traditional derivations of fluctuation theorems, and, in particular, the TasakiCrooks fluctuation theorem [9,11], can be obtained within our framework. Traditionally, one allows an arbitrary unitary operation on the system and heat bath, and performs an energy measurement before and after this unitary. The difference in energy between the initial and final state is taken to be the amount of work extracted or expended. Other derivations assume some particular master equation (e.g., Langevin dynamics), which can be thought of as being generated by some particular family of unitaries. On the other hand, we work with thermal operations (TO), where we explicitly include a work system (a weight) and allow only unitaries that conserve the total energy of system, bath, and weight. Here, we show that the former paradigms are contained in the one we consider here. In order to do this, we take the state of the weight to have coherences over energy levels, which allows us to implement arbitrary unitaries on system and bath. While this is also shown in Ref. [21], here, we clarify a number of issues in the context of fluctuation relations. We note that coherences in the weight are needed only if we wish to explicitly model a unitary that creates coherences over energy levels. However, since fluctuation theorem results typically require that the initial and final state of the system is measured in the energy eigenbasis, we could

0

ASB ðtÞ ¼ eitðHS þHB Þ V SB e−itðHS þHB Þ : Proof.—Most of the following arguments do not exploit the system-bath partition. Hence, in order to simplify the expressions, we jointly call them “composite” C ¼ SB, as in HC ¼ HS þ HB or ρCW ¼ ρSBW. We impose the three fundamental assumptions on the global unitary UCW. We start by imposing the independence of the “position” of the weight. For this, we note that the only operators that commute with ΔW are the functions of itself, fðΔW Þ, and that a complete basis of these functions are the imaginary exponentials eiEΔW . Hence, the condition ½UCW ; ΔW  ¼ 0 implies Z U CW ¼ dEAC ðEÞ ⊗ eiEΔW ; ðA1Þ where AC ðEÞ with E ∈ R is a one-parameter family of operators. Next, we impose energy conservation: UCW ðHC þ H W Þ ¼ ðH 0C þ HW ÞUCW :

ðA2Þ

Note that the equation ½HW ; ΔW  ¼ i implies that ½HW ; eiEΔW  ¼ −EeiEΔW and

041017-10

FLUCTUATING WORK: FROM QUANTUM … Z dE½AC ðEÞHC −

H0C AC ðEÞ

þ EAC ðEÞ ⊗

eiEΔW

PHYS. REV. X 6, 041017 (2016) ¼ 0:

This and the linear independence of the operators eiEΔW gives H0C AC ðEÞ

P If the spectraP of HC and H 0C are discrete, HC ¼ c E c jcihcj and H0C ¼ c0 E c0 jc; ihc; j, then we can write the above as X U CW ¼ jc0 ihc0 jV C jcihcj ⊗ eiðE c0 −E c ÞΔW : ðA12Þ c0 ;c

¼ AC ðEÞðH C þ EÞ;

ðA3Þ

for all E ∈ R. If we translate this equation to Fourier space using 1 AC ðEÞ ¼ 2π

Z

dte−iEt AC ðtÞ;

ðA4Þ

▪ We stress that there is no constraint on V C . This type of unitary was used in the context of thermodynamics in Refs. [21,28]. The above result allows one to obtain an explicit form for the effective map on system-bath (after tracing out the weight): ΓSB ðρSB Þ ¼ trW ðU SBW ρSB ⊗ ρW U†SBW Þ Z ¼ dtASB ðtÞρSB A†SB ðtÞhtjρW jti:

we obtain H0C AC ðtÞ ¼ AC ðtÞHC − i∂ t AC ðtÞ:

ðA5Þ

The solutions of this differential equation are 0

AC ðtÞ ¼ eitHC V C e−itHC ;

ðA6Þ

where V C is arbitrary. Finally, we impose unitarity U CW U †CW ¼ 1CW. That is, Z 1C ⊗ 1W ¼

dE 0 dEAC ðE 0 ÞA†C ðEÞ ⊗ eiðE −EÞΔW : 0

ðA7Þ

Using the linear independence of eiEΔW , we obtain Z

dEAC ðEÞA†C ðE þ EÞ ¼ 1C δðEÞ;

Z U CW ¼ Z ¼

dEdte−iEt AC ðtÞ ⊗ eiEΔW dtAC ðtÞ ⊗ jtihtj:

ðA9Þ

An equivalent form can R be obtained by Rusing the eigenprojectors of HC ¼ dEEPE and H0C ¼ dE 0 E 0 PE 0 . That is, Z UCW ¼ Z ¼

0

dE 0 dEdEdteiðE −E−EÞt ½PE 0 V C PE  ⊗ eiEΔW 0

dE 0 dE½PE 0 V C PE  ⊗ eiðE −EÞΔW 0

¼ eiHC ⊗ΔW AC e−iHC ⊗ΔW :

By noting that htjρW jti is a probability distribution, we see that the reduced map on system and bath ΓSB is a mixture of unitaries (Result 1 in Ref. [55]). Hence, the transformation can never decrease the entropy of system and bath, which guarantees that one cannot pump entropy into the weight, which would be a form of cheating. Equation (A13) also implies that if the initial state of the weight ρW is an eigenstate of ΔW , then the mixture of unitaries has only one term; see Result 10. Result 10.—If the weight is in a maximally coherent state, that is, an eigenstate of ΔW with eigenvalue t0 , the effect on system and bath is an arbitrary unitary, ΓSB ðρSB Þ ¼ ASB ðt0 ÞρSB ASB ðt0 Þ† ;

ðA8Þ

for all E ∈ R. If we translate this equation to Fourier space using Eq. (A4), we get AC ðtÞA†C ðtÞ ¼ 1C , which implies V C V †C ¼ 1C . Substituting Eq. (A6) into Eq. (A1) gives

ðA10Þ ðA11Þ

ðA13Þ

ðA14Þ

where 0

0

0

ASB ðt0 Þ ¼ eit HS þHB V SB e−it HS þHB ;

ðA15Þ

and V SB is as defined in Lemma 1. Proof.—In the particular case of Eq. (A13), where ρW ¼ jt0 iht0 j is an eigenstate of ΔW , so that ΔW jt0 i ¼ t0 jt0 i, the integral is then Z ΓSB ðρSB Þ ¼ dtASB ðtÞρSB A†SB ðtÞδðt − t0 Þ ¼ ASB ðt0 ÞρSB ASB ðt0 Þ† :

ðA16Þ

▪ That is, even though this effective map involves tracing out the weight, the result on system-bath is unitary. In addition, this unitary is totally unconstrained, and, in particular, it need not be energy conserving. A typical R form for this unitary is T exp ½ dtH SB ðtÞ, where T is the time-order operator. In summary, thermal operations with fluctuating work can simulate general unitary transformations that do not

041017-11

ALHAMBRA, MASANES, OPPENHEIM, and PERRY preserve energy. Hence, statements such as Results 1–3 that apply to the first type also apply to the second type. This way we include the operations of the usual derivations of Tasaki-Crooks fluctuation theorems [9,11], where arbitrary unitaries on system and bath are allowed. We now outline how one may derive the analogue of fluctuation relations such as Result 6 in this case, given any unitary dynamics, or mixtures of them. There, the work extracted from system and bath is quantified by measuring their energy before and after the transformation, such that the work takes the form of the random variable E 0 − E, where E is a system plus bath energy associated with projector PE . The conditional distribution PðE 0 jEÞ ¼ trS ½PE 0 ΓSB ðPE ρSB PE Þ plays the key role. It is known that if the map ΓSB is unital [as guaranteed by Eq. (A13)], then the matrix PðE 0 jEÞ is doubly stochastic, and the Jarzynski equality holds, as he−βw i ¼ ¼ ¼

X e−βE 0 eβðE−E Þ PðE 0 jEÞ Z S ZB E;E 0 X e−βE 0 X PðE 0 jEÞ Z Z S B E E0 X e−βE 0 E0

Z0 ¼ S: ZS Z B ZS

ðA17Þ

Other relations, such as the second law equality or Crooks theorem, can be derived analogously too. Essentially, the double stochasticity of PðE 0 jEÞ plays the role of Eq. (7) in the derivations of the fluctuation theorems. Equation (A17) works independently of the state of the weight. In particular, it can be a coherent state jt0 i, which will make the reduced map on the system unitary. Hence, the average of the work extracted from system and bath can be equivalently expressed in terms of (i) measurements on their energy or (ii) shifts in the weight. An important caveat of the results of this Appendix is that they require the weight to be in a coherent state jt0 i. While exactly attaining this state is physically impossible, arbitrarily good approximations are possible in principle, allowing for the implementation of maps arbitrarily close to unitary. Here, we are showing that when one wants to implement arbitrary unitaries, coherence (understood as a thermodynamical resource) is needed. However, as we note in Sec. IV in the main text, implementing a unitary that merely maps energy eigenstates to energy eigenstates requires no such coherence. Finally, in a different direction, there is a further set of operations that we can include within our framework, and, in particular, in Result 4. A large part of the literature on resource theoretic approaches to thermodynamics has been built around a set of operations consisting of sequences of transformations of the energy levels (with an associated work cost) and thermalizations between system and bath. Examples of this are Refs. [23,27,42]. On one hand, the

PHYS. REV. X 6, 041017 (2016) thermalization processes are those for which the work cost vanishes, and consist of a stochastic process (possibly between only two levels) for which we have the following constraint: X Pðs0 jsÞeβðEs0 −Es Þ ¼ 1; ðA18Þ s

which is a particular case of Eq. (7) when we take w ¼ 0 (note that for such thermalization processes the system Hamiltonian remains unchanged). The level transformation processes, on the other hand, consist of a change of Hamiltonian that leaves the populations of the energy levels invariant, ðρ; HS Þ → ðρ; H0S Þ. Hence, these correspond to stochastic matrices of the form Pðs0 ; wjsÞ ¼ δs;s0 δEs −Es0 ;w . It can be easily seen that a process like this satisfies Eq. (24). The values of the work distribution that occur in this process are given by the difference between initial and final energy levels, as expected.

[1] C. Jarzynski, Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78, 2690 (1997). [2] G. E. Crooks, Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences, Phys. Rev. E 60, 2721 (1999). [3] U. Seifert, Stochastic Thermodynamics, Fluctuation Theorems and Molecular Machines, Rep. Prog. Phys. 75, 126001 (2012). [4] U. Seifert, Entropy Production along a Stochastic Trajectory and an Integral Fluctuation Theorem, Phys. Rev. Lett. 95, 040602 (2005). [5] V. Y. Chernyak, M. Chertkov, and C. Jarzynski, PathIntegral Analysis of Fluctuation Theorems for General Langevin Processes, J. Stat. Mech. (2006) P08001. [6] T. Hatano and S.-i. Sasa, Steady-State Thermodynamics of Langevin Systems, Phys. Rev. Lett. 86, 3463 (2001). [7] T. Sagawa and M. Ueda, Generalized Jarzynski Equality under Nonequilibrium Feedback Control, Phys. Rev. Lett. 104, 090602 (2010). [8] B. Schumacher, in Proceedings of the Conference on Quantum Information and Foundations of Thermodynamics, ETH, Zurich. [9] H. Tasaki, Jarzynski Relations for Quantum Systems and Some Applications, arXiv:cond-mat/0009244. [10] M. Campisi, P. Talkner, and P. Hänggi, Fluctuation Theorem for Arbitrary Open Quantum Systems, Phys. Rev. Lett. 102, 210401 (2009). [11] P. Talkner, M. Campisi, and P. Hänggi, Fluctuation Theorems in Driven Open Quantum Systems, J. Stat. Mech. (2009) P02025. [12] M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum Fluctuation Relations: Foundations and Applications, Rev. Mod. Phys. 83, 771 (2011). [13] T. Albash, D. A. Lidar, M. Marvian, and P. Zanardi, Fluctuation Theorems for Quantum Processes, Phys. Rev. E 88, 032146 (2013).

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FLUCTUATING WORK: FROM QUANTUM …

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